Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> 0) (sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4)`
Advertisements
उत्तर
`lim_(x -> 0) [(sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4)] = lim_(x -> 0) [(sqrt(x^2 + 1) - 1) xx ((sqrt(x^2 + 1) + 1))/((sqrt(x^2 + 1) + 1)) xx 1/sqrt(x^2 + 16 - 4) xx (sqrt(x^2 + 16) + 4)/(sqrt(x^2 + 16) + 4)]`
= `lim_(x -> 0) [(x^2 + 1 - 1)/(sqrt(x^2 + 1) + 1) xx (sqrt(x^2 + 16) + 4)/(x^2 + 16 - 16)]`
= `lim_(x -> 0) [x^2/(sqrt(x^2 + 1) + 1) xx (sqrt(x^2 + 16) + 4)/x^2]`
= `lim_(x -> 0) [(sqrt(x^2 + 16) + 4)/(sqrt(x^2 + 1) + 1)]`
= `(sqrt(0^2 + 16) + 4)/(sqrt(0^2 + 1) + 1)`
= `(4 + 4)/(1 + 1)`
`lim_(x -> 0) (sqrt(x^2 + 1) - 1)/(sqrt(x^2 + 16) - 4) = 8/2` = 4
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit:
`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`
Evaluate the following limit :
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
Evaluate the following limit :
`lim_(z -> "a")[((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")]`
In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.
`lim_(x -> 1) (x^2 + x + 1)` = 3
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) sin x/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.001 | 0.01 | 0.1 |
| f(x) | 0.99833 | 0.99998 | 0.99999 | 0.99999 | 0.99998 | 0.99833 |
Sketch the graph of a function f that satisfies the given value:
f(– 2) = 0
f(2) = 0
`lim_(x -> 2) f(x)` = 0
`lim_(x -> 2) f(x)` does not exist.
If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning
Evaluate the following limits:
`lim_(x ->) (x^"m" - 1)/(x^"n" - 1)`, m and n are integers
Show that `lim_("n" -> oo) (1^2 + 2^2 + ... + (3"n")^2)/((1 + 2 + ... + 5"n")(2"n" + 3)) = 9/25`
Evaluate the following limits:
`lim_(x -> oo) (1 + 3/x)^(x + 2)`
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/(sin 5x)`
Evaluate the following limits:
`lim_(x -> 0) (2^x - 3^x)/x`
Evaluate the following limits:
`lim_(x -> 0) (1 - cos^2x)/(x sin2x)`
Evaluate the following limits:
`lim_(x -> 0) (tan x - sin x)/x^3`
Choose the correct alternative:
`lim_(x - oo) sqrt(x^2 - 1)/(2x + 1)` =
Choose the correct alternative:
`lim_(x -> 3) [x]` =
Choose the correct alternative:
`lim_(x -> 0) ("e"^(sin x) - 1)/x` =
`lim_(x -> 0) ((2 + x)^5 - 2)/((2 + x)^3 - 2)` = ______.
The value of `lim_(x→0)(sin(ℓn e^x))^2/((e^(tan^2x) - 1))` is ______.
